Matrix multiplication is an essential part of many different computations within computer systems. For example, matrix multiplication is employed in such applications as two- and three-dimensional computer graphics, computer audio, computer speech and voice synthesizing, computer gaming applications, and computer video. Unfortunately, matrix multiplication operations, particularly those involving large or complex matrices, can be time-consuming to compute. In order to improve performance, therefore, applications may forego complex matrix operations, with a resulting loss of output quality. Hence, if a matrix multiplication operation can be carried out more quickly, better quality results can be produced by various applications using such an operation. For example, graphics can be more detailed and have higher resolution, and graphical processing can involve more complex filters and transitions between window states when faster matrix multiplication is employed. Likewise, audio applications can provide a fuller sound using a wider dynamic range when more complex matrix multiplication associated with producing such audio can be performed in a time-efficient manner. The same is true for speech processing, video gaming applications, and computer video, in that each of these applications benefit from a faster, more efficient matrix multiplication, which allows for a more realistic motion video with higher resolution and faster frame refresh rates.
To this end, it is desirable to be able to perform matrix multiplication in a vector processing system, where an operation can be performed on multiple elements of a matrix with a single instruction. Such a system offers the potential for increased throughput, relative to a scalar processing system in which operation can only be carried out on one element of a matrix at a time. One problem commonly associated with matrix multiplication in a vector processing system, however, is that one of the matrices being multiplied must be transposed. This is due to the manner in which the elements of a matrix must be stored in a data register before a vector operation can be performed. The need to perform such a transposition can require numerous clock cycles, thereby reducing efficiency. A scalar processor is often implemented to perform matrix transposition associated with matrix multiplication. Changing between vector processing and scalar processing engines requires additional numerous clock cycles, and is inefficient compared to processing exclusively within a vector processing system.
Such delays associated with transposing matrices, or with switching between vector processing and scalar processing engines, are exacerbated when large matrices having high dimensions are involved. For example, matrix multiplication of 16×16 matrices each having 16 rows and 16 columns becomes vastly inefficient when performing scalar processing, or when requiring transposition before multiplication. However, many of the aforementioned applications which use matrix multiplication require multiplication of matrices much larger than this. The inefficiencies associated with handling of matrix multiplication by a scalar processor, or transposition of matrices, become greater as the matrix size increases.
Another problem with performing matrix operations in vector processing systems is that rounding errors may be introduced from changes in the order of operations in manipulating the matrix. Such rounding errors are problematic when floating point calculations are carried out and disparities between calculations having different orders of operations become a factor. For example, if a precision is pre-determined, and a calculation is carried out in different sequences, it may be possible that a value used in a later calculation is truncated, thereby yielding a result that varies from a calculation that is carried out on the same two matrices but in a different sequence, or order.
Therefore, it is desirable that a method and system for performing efficient matrix multiplication be devised. It is further desirable that such a system and method for efficient matrix multiplication be suited for performing such tasks within a vector processing system. In this manner, the vector processing system's capabilities may be used in an advantageous manner, thereby increasing the speed and efficiency with which matrices can be multiplied in connection with various computer applications, allowing for improved performance and speed of those applications. It is also desirable that a method and system be devised for matrix multiplication in a manner that is bit-by-bit compatible with the traditional method of matrix multiplication using matrix transposition and/or scalar processing, to prevent discrepancies introduced by inconsistent rounding or changes in order of operations.